arXiv subjects

Carola-Bibiane Schönlieb

Publications and source records attributed to Carola-Bibiane Schönlieb.

At least 19 recordsLinked to original sources

arXiv2026

When is a System Discoverable from Data? Discovery Requires Chaos

The deep learning revolution has enabled major AI advances in physical sciences, particularly in discovering dynamical systems from observational data. Yet the reliability of learned surrogates and symbolically discovered models is often undermined by non-uniqueness: models may fit observed trajectories perfectly, but lack genuine predictive power. We show that chaos, despite its association with unpredictability, is often what makes governing equations discoverable. We prove that the classical Lorenz system is analytically discoverable and verify experimentally that for many systems out-of-domain generalization emerges precisely at chaos onset. Conservative systems, by contrast, are inherently non-discoverable from data alone. The prevalence of chaotic systems in benchmarks may have inadvertently obscured this limitation. These findings help explain why AI excels at weather forecasting, yet struggles in many engineering applications: the systems most amenable to discovery (chaotic ones) are often least amenable to control.

math.DS↗

arXiv2026

Answer-Distribution Trajectories: A Stochastic-Dynamics View of LLM Reasoning

Chain-of-thought reasoning provides a structured computation between a model's input and final answer. Yet it is often evaluated through endpoint accuracy, which ignores the path taken to reach that answer. An emerging line of work addresses this limitation using entropy profiles, which track how uncertainty evolves over the reasoning process but do not reveal which competing hypotheses account for that uncertainty. We introduce answer-distribution trajectories, a stochastic-dynamics-inspired representation that tracks the model's full predictive distribution over answers as reasoning unfolds. As a strictly finer representation than endpoint and entropy summaries, answer-distribution trajectories enable us to characterize a trace through a dynamical reasoning profile spanning exploration, revision, motion, and commitment, and to distinguish different dynamical mechanisms of reasoning success and failure. Across sixteen open-weight language models and four reasoning benchmarks, we show that traces with the same endpoint and similar entropy profiles can exhibit substantially different reasoning dynamics. We further find substantial variation in these dynamics both within and across models and tasks, with different objectives favoring different dynamical profiles. Additionally, we show that training and inference choices systematically reshape these profiles. Our results suggest that answer-distribution trajectories provide a rich framework for analysing and evaluating the dynamics of LLM reasoning.

cs.AI↗

arXiv2026

Geometry-Aware Diffusion Approximate Posterior Sampling for Sparse-View and Limited-Angle CT

Sparse-view computed tomography (CT) reduces radiation dose and acquisition time and may mitigate motion artifacts. However, angular undersampling provides insufficient information to determine the image uniquely and stably. Limited-angle CT, arising from restricted angular coverage, produces strongly directional information loss associated with the missing angular range. In both settings, image directions may be strongly observed, weakly constrained, or unobservable, leading to severe ill-posedness and reconstruction ambiguity. Existing diffusion-based approaches incorporate measurement information through likelihood guidance, data-consistency operations, or range-null-space corrections. However, they do not generally use the continuously varying measurement sensitivity of the acquisition to jointly shape both reconstruction updates and stochastic exploration. We propose a geometry-aware diffusion-guided stochastic reconstruction framework for sparse-view and limited-angle CT. Its central component is a regularized noise-weighted pullback metric constructed from the CT forward operator and measurement-noise covariance. This metric continuously adapts both the measurement-aware update and stochastic exploration according to directional measurement sensitivity, suppressing changes along strongly constrained directions while permitting greater exploration along weakly constrained and unobservable directions. We complement this geometry-aware update with a regularized data-consistency correction and approximately null-space-restricted stochastic perturbations, implemented matrix-free using forward and backprojection operations together with conjugate-gradient solves. Experiments on sparse-view, noisy, and limited-angle CT demonstrate competitive reconstruction quality, strong measurement consistency, and spatially resolved empirical uncertainty estimates.

eess.IV↗

arXiv2026

Trajectory Stitching for Solving Inverse Problems with Flow-Based Models

Flow-based generative models have emerged as powerful priors for solving inverse problems. One option is to directly optimize the initial latent code (noise), such that the flow output solves the inverse problem. However, this requires backpropagating through the entire generative trajectory, incurring high memory costs and numerical instability. We propose MS-Flow, which represents the trajectory as a sequence of intermediate latent states rather than a single initial code. By enforcing the flow dynamics locally and coupling segments through trajectory-matching penalties, MS-Flow alternates between updating intermediate latent states and enforcing consistency with observed data. This reduces memory consumption while improving reconstruction quality. We demonstrate the effectiveness of MS-Flow over existing methods on image recovery and inverse problems, including inpainting, super-resolution, and computed tomography.

eess.IV↗

arXiv2026

Beyond Unimodal Bases: Pullback Geometry for Multimodal Data

Data-driven Riemannian geometry provides nonlinear interpolation and geometric representations of high-dimensional data. For these operations to be statistically meaningful, paths between observations should preferentially traverse high-likelihood regions. Existing scalable pullback constructions typically use a unimodal Gaussian latent distribution, assuming that the data reside close to a single manifold. For multimodal data, mapping separated modes or local structures into one Gaussian region can require substantial transport deformation and compromise the resulting geometry. We introduce a pullback geometry for data supported on mixtures of manifolds. Using a latent Gaussian mixture, we define its Riemannian metric as the matrix square of the responsibility-weighted expected component precision. The metric is smooth and positive definite and recovers the existing Gaussian construction in the single-component limit. For structured overlapping mixtures, we establish conditions under which the log-density is concave along geodesics, providing a formal connection between the proposed geometry and paths through high-likelihood regions, and derive the corresponding local curvature relations. We instantiate this geometry in a normalizing flow with adaptive mixture learning, allowing the number of active components to emerge from the data and supporting component-wise reconstruction and local effective-dimension estimation. Experiments on synthetic geometric data, a controlled multi-view image setting with a known reference trajectory, and MNIST show reduced transport distortion, competitive path support, close reference-trajectory recovery, and improved interpolation realism. These results extend scalable pullback geometry beyond datasets that reside close to a single manifold while retaining tractable and interpretable local structure.

cs.LG↗

arXiv2026

GRIFDIR: Graph Resolution-Invariant Diffusion Models over Irregular Domains

Score-based diffusion models in infinite-dimensional function spaces provide a mathematically principled framework for modelling function-valued data, offering key advantages such as resolution invariance and the ability to handle irregular discretisations. However, practical implementations have struggled to fully realise these benefits. Existing backbones like Fourier neural operators are often biased towards regular grids and fail to generalise to complex domain topologies. We introduce an architecture, GRIFDIR, for function-space diffusion models that represents generalised graph convolutional kernels as finite element functions, allowing the score network to operate directly on unstructured meshes over domains of arbitrary shape. We demonstrate the efficacy of our network architecture through a series of unconditional and conditional sampling experiments across diverse geometries, including non-convex and multiply-connected domains. Our results show that the proposed method maintains resolution invariance and achieves high fidelity in capturing functional distributions on non-trivial geometries.

cs.LG↗

arXiv2026

Mixed-Noise Plug-and-Play with Infimal Convolution Fidelities and Multiple Priors

Plug-and-Play (PnP) algorithms are a class of iterative methods for inverse imaging. Within an optimization algorithm, they combine a flexible fidelity term, encoding the forward operator, and a pretrained image denoiser, in order to deal with more severe corruptions such as blurring or downsampling when reconstructing an image. This work studies provably convergent PnP methods for mixed-noise forward processes by using the infimal convolution as a fidelity term, providing a statistical interpretation as a joint maximum a-posteriori estimator over both noises, and preserving the Bayesian MAP interpretation of PnP methods. Independently, we extend the PnP formulation to multiple prior terms using the Davis--Yin three-operator splitting. This extension can be combined with either standard fidelities or the proposed mixed-noise infimal-convolution fidelities. We verify that these generalized PnP methods are convergent under standard Kurdyka--Lojasiewicz conditions. Numerical experiments on Laplace-Gaussian and Poisson--Gaussian noise demonstrate stable convergence of single-prior and multiple-prior PnP methods, and divergence under fidelity mismatch. Furthermore, PnP with infimal convolution fidelities are able to scale to noise up to 38% standard deviation, with multiple priors reaching different stationary points that qualitatively preserve more textural properties.

math.OC↗

arXiv2026

Learning piecewise-smooth dynamical systems

Discovering dynamical systems from trajectory data is a central problem in applied mathematics and engineering. Whilst recent advances in machine learning have led to strong progress in data-driven system identification, much less attention has been given to systems with discontinuous dynamics. These systems are nevertheless highly relevant in applications, including climate dynamics and mechanical systems with friction. In this work, we consider the problem of identifying piecewise-smooth dynamical systems directly from trajectory data. Compared with the smooth setting, this requires recovering the governing equations and detecting the switching hyperplanes that separate different dynamical regimes and characterising their behaviour, such as sliding motion. We present a modular framework for discovering such systems by first estimating switching hyperplanes from data and then learning smooth dynamics within each region using geometry-constrained neural networks. The geometry-learning phase is studied from a statistical perspective, analysing the identifiability of the discontinuities and the robustness of the procedure. We also introduce a novel neural network architecture with a prescribed discontinuity set, and provide a theoretical analysis of its approximation properties. The approach is tested on low-dimensional benchmark problems, including dry-friction oscillators and the PP04 climate model for the ice ages.

math.DS↗

arXiv2026

Recovering protein conformations from single-particle cryo-EM data via indirect shape matching gradient flows

Single-particle cryo-electron microscopy images a macromolecule as many noisy tomographic projections of its electrostatic potential. We reconstruct the protein backbone directly from such projections, as an atomic point cloud, without the intermediate step of reconstructing the 3D electrostatic potential map. We formulate this as an indirect shape matching problem: a point-cloud template of the backbone is deformed until its simulated projections agree with the data, with the structure observed only through the imaging operator. The deformation is computed via a gradient flow on a Lie group, and we derive the resulting framework in a general geometric setting before adapting it for single-particle cryo-electron microscopy. On synthetic data, we recover single- and multichain proteins and capture conformational transitions.

q-bio.BM↗

arXiv2026

1-Lipschitz Neural Networks on Hadamard Manifolds

Controlling the Lipschitz constant of a neural network is a standard way to promote robustness and stability. Most existing constraining strategies are designed for Euclidean spaces. In this work, we construct and analyze a class of 1-Lipschitz neural networks on Hadamard manifolds. Our layers are of gradient-descent type, $1$-Lipschitz, and quasi-$α$-firmly nonexpansive. The core building blocks of the proposed architecture are Busemann functions, and we exploit the properties of Busemann gradient flows to design $1$-Lipschitz geometry-preserving layers. We provide explicit constructions and examples for hyperbolic manifolds and the manifold of symmetric positive definite (SPD) matrices. We test the proposed architecture in two numerical experiments: robust classification on the Poincaré disk and masked-Wishart covariance reconstruction. On the Poincaré disk, the proposed networks yield robust classifiers under hyperbolic perturbations. On the SPD manifold, we train SPD-valued denoisers and adopt them as a Plug-and-Play prior for a masked-Wishart covariance reconstruction problem. We show improved results from the nonexpansive denoiser over static, data-only, and Log-Euclidean denoising baselines, and empirically test its convergence properties.

math.NA↗

arXiv2026

CBCT-IQ: A Publicly Available Annotated Cone-Beam CT Dataset for Image Quality Assessment and Benchmarking

Medical image quality plays a critical role in diagnostic accuracy, especially in X-ray-based imaging modalities such as cone-beam computed tomography (CBCT), where image quality must be balanced against radiation dose. While expert visual evaluation remains the clinical standard for image quality evaluation, it is time-consuming, subjective and affected by inter-observer variability, emphasizing the need for reliable quantitative image quality assessment (IQA) methods. However, the development and validation of such IQA methods have been limited by the lack of publicly available CBCT datasets with expert image quality annotations. In this study, we provide the first open-access CBCT IQA dataset containing 1,764 annotated image slices acquired using systematic variations in image acquisition and reconstruction parameters. Three clinical experts graded the overall image quality and a predefined regions of interest (ROI) using a four-level scoring scheme. In addition, we benchmark 26 full reference- and no reference-based IQA measures against expert annotations and introduce an exploratory IQA measure-based ranking capable of distinguishing subtle image quality differences. This dataset introduced a standardized benchmark for future CBCT IQA research and provides a valuable resource for the development and validation of new IQA methods, enabling reproducible research and advancing CBCT IQA.

physics.med-ph↗

arXiv2026

Rethinking Clinical Relevance in Chest X-ray Machine Learning: How Evaluation References Define Performance

Chest X-ray (CXR) machine learning relies heavily on automated evaluation using reference standards that aim to approximate clinical judgment. However, commonly used report-derived labels for pathology classification or generic image quality metrics for reconstruction may not reliably reflect clinical judgment. We systematically investigate how evaluation-reference choices affect model performance and ranking in both pathology classification and image quality assessment (IQA). To enable controlled comparison across evaluation references, we collected paired expert image- and report-derived labels for thoracic findings from a clinical cohort at Cambridge University Hospitals (CUH) and curated a subset of the public MIMIC-CXR dataset, along with expert ratings of diagnostic image quality. We show that for supervised image classifiers (ResNet, DenseNet), several zero-shot and fine-tuned vision-language models (e.g., MedKLIP, GLoRIA, and ConVIRT), changing the label source leads to substantial differences not only in performance estimates but also in model rankings. In parallel, alignment of IQA measures with expert judgment depends heavily on the choice of measure, and commonly used IQA metrics such as SSIM and PSNR often fail to align with expert assessments of diagnostic usability. Our results demonstrate that evaluation choices are crucial: they can determine which models and methods appear best and are therefore selected for further development or deployment. The selection of evaluation references should therefore be treated as a central component of clinical validity in CXR machine learning, and justified with respect to the pathology, imaging task, and intended downstream clinical use.

eess.IV↗

arXiv2026

DeepCormack: Fermi surface tomography using model-based data-driven algorithms

The experimental reconstruction of the 3D two-photon momentum density (TPMD) via angular correlation of electron-positron annihilation radiation (ACAR) is a particularly useful method for studying material Fermi surfaces. It does not rely on low temperatures, UHV conditions, or strong magnetic fields, and enables the study of the spin-resolved electronic structure of materials. Yet, it remains a challenging inverse problem. Typically, 10^8 positron annihilation events are measured for 3--6 projections of the TPMD at different angles. The standard reconstruction approach is an ACAR adaptation of Cormack's method (the MCM) that leverages the inherent symmetry in the crystal's structure. However, the poor signal-to-noise ratio means collecting data of sufficient quality for Fermi surface studies can take months per sample. We present DeepCormack, a family of data-driven model-based reconstruction algorithms that augments the MCM by integrating supervised deep-learning models (CNN, MLP, and UNet) at various stages. To overcome the lack of large experimental training sets, we propose a method which leverages singular value decomposition with dynamic mode decomposition to generate realistic synthetic TPMD volumes, requiring only a single reference momentum density computed via density functional theory. On test data, DeepCormack improves reconstruction quality over MCM by about 8.5 dB PSNR at 200M counts and remains stable at reduced counts, enabling significantly faster acquisition times. Generalisation to experimental data depends strongly on how well the training distribution from the reference momentum density matches the sample. We therefore recommend pairing DeepCormack with a DFT calculation of the target material to create sample-specific training data. Our proposed method offers either much higher quality reconstructions, or enables significantly faster ones, on the order of weeks.

cond-mat.mtrl-sci↗

arXiv2026

FLASH: Flexible Learning of Adaptive Sampling from History in Temporal Graph Neural Networks

Aggregating temporal signals from historic interactions is a key step in future link prediction on dynamic graphs. However, incorporating long histories is resource-intensive. Hence, temporal graph neural networks (TGNNs) often rely on historical neighbors sampling heuristics such as uniform sampling or recent neighbors selection. These heuristics are static and fail to adapt to the underlying graph structure. We introduce FLASH, a learnable and graph-adaptive neighborhood selection mechanism that generalizes existing heuristics. FLASH integrates seamlessly into TGNNs and is trained end-to-end using a self-supervised ranking loss. We provide theoretical evidence that commonly used heuristics hinder TGNNs performance, motivating our design. Extensive experiments across multiple benchmarks demonstrate consistent and significant performance improvements for TGNNs equipped with FLASH.

cs.LG↗

arXiv2026

Multi-Headed Transformer Architectures as Time-dependent Wasserstein Gradient Flows

In recent years, transformer architectures have revolutionized the field of language processing, opening the door to previously unforeseen possibilities. However, from a theoretical point of view, the mathematical models proposed in the literature often lack direct contact with the actual architectures and depend on strong simplifying assumptions. In this paper, we reduce this gap by modelling the data flow in multi-headed transformer architectures as time-dependent gradient flows for a suitable interaction energy capturing the design of the attention mechanism. The explicit dependence on time allows us to consider different weights for each head and for each layer, without imposing constraints on the initialization method. Moreover, we prove that, under a suitable integrability assumption on the evolution of the weights, each element of the $ω$-limit set of the gradient flows is a stationary point of the interaction energy at a limiting weight distribution. Finally, we analyse the stability of the gradient flows considering perturbations of both the initial data and the weights. Specifically, on the one hand, we study the robustness of the proposed models with respect to noisy inputs, establishing a continuous dependence of the gradient flows on the initial data and uniqueness of the flows. On the other hand, we prove the $Γ$-convergence of the perturbed interaction energy to the unperturbed one, leading to the convergence of the corresponding gradient flows. We complement these theoretical results with numerical experiments that confirm the predicted energy-dissipation identity and clarify the asymptotic behavior of the dynamics in both the autonomous-like (Ornstein--Uhlenbeck) and the genuinely non-autonomous (oscillating-weights) regimes.

cs.LG↗

arXiv2026

Muon is Not That Special: Random or Inverted Spectra Work Just as Well

The recent empirical success of the Muon optimizer has renewed interest in non-Euclidean optimization, typically justified by similarities with second-order methods, and linear minimization oracle (LMO) theory. In this paper, we challenge this geometric narrative through three contributions, demonstrating that precise geometric structure is not the key factor affecting optimization performance. First, we introduce Freon, a family of optimizers based on Schatten (quasi-)norms, powered by a novel, provably optimal QDWH-based iterative approximation. Freon naturally interpolates between SGD and Muon, while smoothly extrapolating into the quasi-norm regime. Empirically, the best-performing Schatten parameters for GPT-2 lie strictly within the quasi-norm regime, and thus cannot be represented by any unitarily invariant LMO. Second, noting that Freon performs well across a wide range of exponents, we introduce Kaon, an absurd optimizer that replaces singular values with random noise. Despite lacking any coherent geometric structure, Kaon matches Muon's performance and retains classical convergence guarantees, proving that strict adherence to a precise geometry is practically irrelevant. Third, having shown that geometry is not the primary driver of performance, we demonstrate it is instead controlled by two local quantities: alignment and descent potential. Ultimately, each optimizer must tune its step size around these two quantities. While their dynamics are difficult to predict a-priori, evaluating them within a stochastic random feature model yields a precise insight: Muon succeeds not by tracking an ideal global geometry, but by guaranteeing step-size optimality.

cs.LG↗

arXiv2026

CATO: Charted Attention for Neural PDE Operators

Neural operators have emerged as powerful data-driven solvers for PDEs, offering substantial acceleration over classical numerical methods. However, existing transformer-based operators still face critical challenges when modeling PDEs on complex geometries: directly processing over massive mesh points is computationally expensive, while operating in raw discretization coordinates may obscure the intrinsic geometry where physical interactions are more naturally expressed. To address these limitations, we introduce the Charted Axial Transformer Operator (CATO), a geometry-adaptive and derivative-aware neural operator for PDEs on general geometries. Instead of applying attention directly in the physical coordinate system, CATO learns a continuous latent chart that maps mesh coordinates into a learned chart space, where chart-conditioned axial attention efficiently captures long-range dependencies with reduced computational cost. In addition, CATO introduces a derivative-aware physics loss for steady-state PDEs that jointly supervises solution values, mesh-consistent gradients, and an auxiliary flux-like field, improving physical fidelity and reducing oversmoothing. We further provide a theoretical approximation result showing that, under a favorable chart, charted axial attention can represent low-rank axial solution operators with controlled error, and that small chart perturbations induce bounded approximation degradation. CATO achieves the best performance across all evaluated datasets, yielding an average improvement of approximately 26.76\% over the strongest competing baselines while reducing the number of parameters by 81.98\%. These results highlight the effectiveness of learning geometry-adaptive charts and derivative-aware physical supervision for accurate and efficient PDE operator learning.

cs.AI↗

arXiv2026

Expressivity of Bi-Lipschitz Normalizing Flows: A Score-Based Diffusion Perspective

Many normalizing flow architectures impose regularity constraints, yet their distributional approximation properties are not fully characterized. We study the expressivity of bi-Lipschitz normalizing flows through the lens of score-based diffusion models. For the probability flow ODE of a variance-preserving diffusion, Lipschitz regularity of the score induces a flow of bi-Lipschitz diffeomorphic transport maps. This ODE bridge allows us to analyze the distributional approximation power of bi-Lipschitz normalizing flows and, conversely, derive deterministic convergence guarantees for diffusion-based transport. Our key idea is to use the probability flow ODE to link regularity of the score to regularity of the induced transport maps. We verify score regularity for broad target densities, including compactly supported densities, Gaussian convolutions of compactly supported measures and finite Gaussian mixtures. We obtain a universal distributional approximation result: Gaussian pullbacks induced by bi-Lipschitz variance-preserving transport maps are $L^1$-dense among all probability densities. For Gaussian convolution targets, we further obtain convergence in Kullback-Leibler divergence without early stopping.

stat.ML↗